Question: $ 0.\overline{1} \div 0.\overline{66} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 10x &= 1.1111...\\ x &= 0.1111...\end{align*} $ $\begin{align*} 9x &= 1 \\ x &= \dfrac{1}{9}\end{align*} $ $\begin{align*} 100y &= 66.6666...\\ y &= 0.6666...\end{align*} $ $\begin{align*} 99y &= 66 \\ y &= \dfrac{66}{99}\end{align*} $ So, the problem becomes: $ \dfrac{1}{9} \div \dfrac{66}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{1}{9} \times \dfrac{99}{66} = {?} $ $ \phantom{\dfrac{1}{9} \times \dfrac{66}{99}} = \dfrac{1 \times 99}{9 \times 66} $ $ \phantom{\dfrac{1}{9} \times \dfrac{66}{99}} = \dfrac{1 \times \cancel{99}11} {\cancel{9} \times 66} $ $ \phantom{\dfrac{1}{9} \times \dfrac{66}{99}} = \dfrac{11}{66} $ Simplify: ${= \dfrac{1}{6}}$